Coulomb effects in granular materials at not very low temperatures

We consider effects of Coulomb interaction in a granular normal metal at not very low temperatures suppressing weak localization effects. In this limit calculations with the initial electron Hamiltonian are reduced to integrations over a phase variable with an effective action, which can be considered as a bosonization for the granular metal. Conditions of the applicability of the effective action are considered in detail and importance of winding numbers for the phase variables is emphasized. Explicit calculations are carried out for the conductivity and the tunneling density of states in the limits of large $g\gg 1$ and small $g\ll 1$ tunnelling conductances. It is demonstrated for any dimension of the array of the grains that at small $g$ the conductivity and the tunnelling density of states decay with temperature exponentially. At large $g$ the conductivity also decays with decreasing the temparature and its temperature dependence is logarithmic independent of dimensionality and presence of a magnetic field. The tunnelling density of states for $g\gg 1$ is anomalous in any dimension but the anomaly is stronger than logarithmic in low dimensions and is similar to that for disordered systems. The formulae derived are compared with existing experiments. The logarithmic behavior of the conductivity at large $g$ obtained in our model can explain numerous experiments on systems with a granular structure including some high $T_{c}$ materials.